Entire Solutions Of Certain Type Research Articles

Some results on the transcendental entire solutions of certain type of non-linear shift-differential equations

Some results on the transcendental entire solutions of certain type of non-linear shift-differential equations

The Exact Entire Solutions of Certain Type of Nonlinear Difference Equations

In this paper, we consider the entire solutions of nonlinear difference equation $$f^{3}+q(z)\Delta f=p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}$$ , where $$q$$ is a polynomial, and $$p_{1},p_{2},\alpha_{1},$$ and $$\alpha_{2}$$ are nonzero constants with $$\alpha_{1}\neq\alpha_{2}$$ . It is showed that if $$f$$ is a nonconstant entire solution of $$\rho_{2}(f)<1$$ to the above equation, then $$f(z)=e_{1}e^{\frac{\alpha_{1}z}{3}}+e_{2}e^{\frac{\alpha_{2}z}{3}}$$ , where $$e_{1}$$ and $$e_{2}$$ are two constants. Meanwhile, we give an affirmative answer to the conjecture posed by Zhang et al in [18].

一类微分–差分方程的整函数解

一类微分–差分方程的整函数解

Entire solutions of certain type of non-linear differential equations

AbstractUtilizing Nevanlinna’s value distribution theory of meromorphic functions, we study transcendental entire solutions of the following type nonlinear differential equations in the complex plane$$\begin{array}{} \displaystyle f^{n}(z)+P(z,f,f',\ldots,f^{(t)})=P_{1}\text{e}^{\alpha_{1}z}+P_{2}\text{e}^{\alpha_{2}z}+P_{3}\text{e}^{\alpha_{3}z}, \end{array}$$where Pj and αi are nonzero constants for j = 1, 2, 3, such that |α1| &gt; |α2| &gt; |α3| and P(z, f, f′, …, f(t) is an algebraic differential polynomial in f(z) of degree no greater than n – 1.

On entire solutions of certain type of nonlinear differential equations

In this paper, we shall extend some results regarding the growth estimate of entire solutions of certain type of linear differential equations to that of nonlinear differential equations. Moreover, our results will include several known results for linear differential equations obtained earlier as special cases.

Entire Solutions of Certain Type of Non-Linear Difference Equations

In this paper, we study the existence of entire solutions of finite-order of non-linear difference equations of the form $$\begin{aligned} f^{n}(z)+q(z)\Delta _{c}f(z)=p_{1}\mathrm{e}^{\alpha _{1}z}+p_{2}\mathrm{e}^{\alpha _{2}z},\quad n\ge 2 \end{aligned}$$ and $$\begin{aligned} f^{n}(z)+q(z)\mathrm{e}^{Q(z)}f(z+c)=p_{1}\mathrm{e}^{\lambda z}+p_{2}\mathrm{e}^{-\lambda z},\quad n\ge 3 \end{aligned}$$ where q, Q are non-zero polynomials, $$c,\lambda ,p_{i},\alpha _{i}(i=1,2)$$ are non-zero constants such that $$\alpha _{1}\ne \alpha _{2}$$ and $$\Delta _{c}f(z)=f(z+c)-f(z)\not \equiv 0$$ . Our results are improvements and complements of Wen et al. (Acta Math Sin 28:1295–1306, 2012), Yang and Laine (Proc Jpn Acad Ser A Math Sci 86:10–14, 2010) and Zinelâabidine (Mediterr J Math 14:1–16, 2017).

On Properties of Entire Solutions of Difference Equations and Difference Polynomials

Abstract In this paper, we consider the existence of entire solutions of certain type of non-linear difference equation of the form: f(z)n + p(z)f(z + c)m = q(z). We also consider value distribution problems of difference polynomials of entire functions such as f(z)n(f(z)−1)f(z +c) for n = 1, which is a supplement of previous results.

ON ENTIRE SOLUTIONS OF CERTAIN TYPE OF DIFFERENTIAL-DIFFERENCE EQUATIONS

In this paper, we deal with differential-difference equations of the form $$ f(z)^2+p(z)f(z+c)+h(z)f'(z)+g(z)=d_1e^{\lambda z}+d_2e^{-\lambda z} $$ where $p(z),~ h(z),~ g(z)$ are polynomials, and $c,~ d_1,~d_2, ~\lambda\in \mathbb{C}$ are constants with $d_1 d_2\lambda\not= 0$. By utilizing Nevanlinna's value distribution theory, some sufficient conditions on the nonexistence of entire solutions regarding the equations are provided.

Entire solutions of certain type of difference equations

Abstract In this paper, we shall study the conditions regarding the existence of transcendental entire solutions of certain type of difference equations. Our results are either supplements to some results obtained recently, or are relating to the conjecture raised in Yang and Laine (Proc. Jpn. Acad., Ser. A, Math. Sci. 86:10-14, 2010). Finally, two relevant conjectures are posed for further studies. MSC:39B32, 34M05, 30D35.

Uniqueness and value distribution of differences of entire functions

We consider the existence of transcendental entire solutions of certain type of non-linear difference equations. As an application, we investigate the value distribution of difference polynomials of entire functions. In particular, we are interested in the existence of zeros of fn(z)(λfm(z+c)+μfm(z))−a, where f is an entire function, n, m are two integers such that n⩾m>0, and λ, μ are non-zero complex numbers. We also obtain a uniqueness result in the case where shifts of two entire functions share a small function.

Entire solutions of certain type of differential equations II

We analyze the transcendental entire solutions of the following type of nonlinear differential equations: f n ( z ) + P ( f ) = p 1 e α 1 z + p 2 e α 2 z in the complex plane, where p 1 , p 2 and α 1 , α 2 are nonzero constants, and P ( f ) denotes a differential polynomial in f of degree at most n − 1 with small functions of f as the coefficients.

Entire solutions of certain type of differential equations

By utilizing Nevanlinna's value distribution theory of meromorphic functions, we solve the transcendental entire solutions of the following type of nonlinear differential equations in the complex plane: f n ( z ) + P ( f ) = p 1 e α 1 z + p 2 e α 2 z , where p 1 and p 2 are two small functions of e z , and α 1 , α 2 are two nonzero constants with some additional conditions, and P ( f ) denotes a differential polynomial in f and its derivatives (with small functions of f as the coefficients) of degree no greater than n − 1 .

On the nonexistence of entire solutions of certain type of nonlinear differential equations

By utilizing Nevanlinna's value distribution theory of meromorphic functions, it is shown that the following type of nonlinear differential equations: f n ( z ) + P n − 3 ( f ) = p 1 e α 1 z + p 2 e α 2 z has no nonconstant entire solutions, where n is an integer ⩾4, p 1 and p 2 are two polynomials ( ≢ 0 ) , α 1 , α 2 are two nonzero constants with α 1 / α 2 ≠ rational number, and P n − 3 ( f ) denotes a differential polynomial in f and its derivatives (with polynomials in z as the coefficients) of degree no greater than n − 3 . It is conjectured that the conclusion remains to be valid when P n − 3 ( f ) is replaced by P n − 1 ( f ) or P n − 2 ( f ) .